Mathematics > Differential Geometry

Title:
Dynamics of the Automorphism Group of the GL(2,R)-Characters of a Once-puncutred Torus

Abstract: Let pi be a free group of rank 2. Its outer automorphism group Out(pi) acts
on the space of equivalence classes of representations in Hom(pi, SL(2,C)). Let
SLm(2,R) denote ths subset of GL(2,R) consisting of matrices of determinant -1
and let ISL(2,R) denote the subgroup (SL(2,R) union i SLm(2,R)) of SL(2,C). The
representation space Hom(pi, ISL(2,R)) has four connected components, three of
which consist of representations that send at least on generator of pi to
iSLm(2,R). We investigate the dynamics of the Out(pi)-action on these
components.
The group Out(pi) is commensurable with the group Gamma of automorphisms of
the polynomial kappa(x,y,z) = -x^2 - y^2 + z^2 + xyz -2. We show that for -14 <
c < 2, the action of Gamma is ergodic on the level sets kappa^(-1)(c). For c <
-14 the group Gamma acts properly and freely on an open subset OmegaMc of
kappa^(-1)(c) and acts ergodically on the complement of OmegaMc. We construct
an algorithm which determines, in polynomial time, if a point (x,y,z) in R^3 is
Gamma-equivalent to a point in OmegaMc or in its complement.
Conjugacy classes of ISL(2,R)-representations identify with R^3 via an
appropriate restriction of the Fricke character map. Corresponding to the
Fricke spaces of the once-punctures Klein bottle and the once-punctured Moebius
band are Gamma-invariant open subsets OmegaK and OmegaM respectively. We give
an explicit parametrization of OmegaK and OmegaM as subsets of R^3 and we show
that OmegaM has a non-empty intersection with kappa^(-1)(c) if and only if
c<-14, while OmegaK has a non-empty intersection with kappa^(-1)(c) if and only
if c>6.